18.4 Electric Potential

Electric potential energy and gravitational potential energy follow the same core idea: both store energy based on an object's position in a field. Understanding how they compare (and where they differ) makes electric potential much easier to grasp.

Electric potential, measured in volts, tells you the energy per unit charge at a point in an electric field. It's what determines how charges move: positive charges head toward low potential, negative charges toward high potential.

Electric Potential Energy and Gravitational Potential Energy

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Electric vs gravitational potential energy

Electric potential energy ($U_E$) and gravitational potential energy ($U_G$) both describe energy stored because of an object's position in a field. Comparing them side by side is one of the best ways to build intuition for electric potential energy.

Similarities:

• Both are scalar quantities measured in joules (J)
• Both depend on position relative to a reference point
• Both can convert into kinetic energy as the object moves under the field's influence
• Both involve conservative forces, so the work done depends only on start and end points, not the path taken

Differences:

• $U_E$ comes from the electric force between charges; $U_G$ comes from the gravitational force between masses
• $U_G$ is always negative because gravity is always attractive. $U_E$ can be positive or negative, depending on whether the charges repel (like charges) or attract (opposite charges)
• The formulas look similar but aren't identical:

$U_E = \frac{kq_1q_2}{r}$

where $k = 8.99 \times 10^9 \text{ N·m}^2/\text{C}^2$, $q_1$ and $q_2$ are the charges, and $r$ is the distance between them.

$U_G = -\frac{GMm}{r}$

where $G = 6.67 \times 10^{-11} \text{ N·m}^2/\text{kg}^2$, $M$ and $m$ are the masses, and $r$ is the distance between their centers.

Notice the negative sign baked into $U_G$. For $U_E$, the sign comes naturally from the charges themselves: two positive charges give positive $U_E$ (repulsion stores energy), while a positive and negative charge give negative $U_E$ (you'd have to do work to pull them apart).

Electric Potential and Electric Fields

Calculation of electric potential difference

Electric potential ($V$) is the electric potential energy per unit charge at a point in an electric field:

$V = \frac{U_E}{q}$

This is a property of the field itself, not of any particular charge you place in it.

Electric potential difference ($\Delta V$) is the change in potential between two points:

$\Delta V = V_f - V_i$

For a point charge, the potential at distance $r$ is:

$V = \frac{kq}{r}$

The potential difference between two distances $r_1$ and $r_2$ from a point charge $q$ is:

$\Delta V = kq\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$

Example: Find the potential difference between points 2 m and 4 m from a $+5 \text{ nC}$ charge.

$\Delta V = (8.99 \times 10^9)(5 \times 10^{-9})\left(\frac{1}{2} - \frac{1}{4}\right) = 11.2 \text{ V}$

In a uniform electric field, the relationship simplifies. For two points separated by distance $d$ along the field direction:

$\Delta V = -Ed$

The negative sign means potential decreases in the direction the field points.

Example: Two parallel plates are 0.05 m apart in a uniform 100 N/C field. The potential difference is:

$\Delta V = -(100)(0.05) = -5 \text{ V}$

Relation of potential to charge behavior

Electric potential tells you how much energy per unit charge is available to do work at a given location. The key rule for how charges respond:

• Positive charges move from high potential to low potential (similar to how masses fall from high to low gravitational potential)
• Negative charges move from low potential to high potential (the opposite direction)

Electric field lines always point in the direction of decreasing potential. That's the direction a free positive charge would accelerate.

The work done by the electric field on a charge $q$ moving through a potential difference is:

$W = q\Delta V$

• A positive charge moving from high to low potential: the field does positive work (the charge speeds up)
• A positive charge moving from low to high potential: the field does negative work (the charge slows down)

The electric field and potential are connected by a gradient relationship:

$\vec{E} = -\nabla V$

In plain terms, the electric field points in the direction where potential drops most steeply, and the field's magnitude equals the rate of that drop. In one dimension, this simplifies to $E = -\frac{\Delta V}{\Delta x}$.

Equipotential Surfaces and Electrostatic Shielding

An equipotential surface is any surface where every point has the same electric potential. Two important properties follow from this:

• Electric field lines are always perpendicular to equipotential surfaces. If they weren't, there'd be a component of the field along the surface, which would mean the potential changes along it.
• No work is done moving a charge along an equipotential surface, since $W = q\Delta V$ and $\Delta V = 0$.

Electrostatic shielding happens when a conductor reaches electrostatic equilibrium. All excess charge sits on the outer surface, and the electric field inside drops to zero. The entire interior (and surface) of the conductor becomes one equipotential region. This is the principle behind a Faraday cage: a conducting enclosure that blocks external electric fields from reaching whatever is inside.

Capacitance measures how much charge a conductor system can store per volt of potential difference:

$C = \frac{Q}{V}$

The unit is the farad (F), where $1 \text{ F} = 1 \text{ C/V}$. One farad is enormous; most real capacitors are measured in microfarads ($\mu\text{F}$) or picofarads ($\text{pF}$).